Real Mathematics – Killer Numbers#6

Socrates’ Lesson

In the previous articles I have talked about Plato and his effect on science; particularly geometry. Thanks to his book named Meno, we know about one of the most influential philosophers of all times: Socrates.


Meno was another book of Plato that was written as dialogues. In this book there were two main characters: Meno and Socrates.

In the beginning of the book Meno asks Socrates if virtue is teachable or not. Even though Meno is crucial for understanding Socrates’ philosophy, there is one part of the book that interests me the most.


The book gets interesting when Socrates starts asking “the boy” who was raised near Meno. At first, Socrates is asking the boy to describe shape of a square and its properties. After a series of questions Socrates asks his main problem: How can one double the area of a given square?

This is an ancient problem that is also known as “doubling the square”. The boy answers Socrates’ questions and eventually finds the area of a square with side length of 2 units. The boy also concludes that since this area has 4 units, double of such square should have 8 units. But when asked to find one side of such square, the boy gives the answer of 4 units. However after his answer the boy realizes that a square with sides of 4 units has 16 units of area, not 8.

Classical Greek Mathematics

After this point the boy follows Socrates’ descriptions in order to draw a square that has 8 units of area. At first Socrates commands the boy to draw a square that has sides 2:


This square’s area is 4 units. Then Socrates tells him to draw three identical squares:


Now Socrates tells the boy to unite these squares as follows:


Socrates asks the boy to draw the diagonals in each square. They both know the fact that a diagonal divides a square into two equal areas:


It is easy to see that the inner square has a total area of 8 units:


One side of the inner square is the diagonal from small squares. In order to find that diagonal the boy uses Pythagorean Theorem:



Even though he only uses a compass and an unmarked ruler, the boy found a length that is irrational thanks to Socrates’ instructions. Back in ancient Greece numbers were imagined as lengths/magnitudes. This is why as long as they constructed it neither Socrates nor the boy cared about irrationality of a length.

Pythagoras and his cult claimed that all numbers are rational and they tried to hide the facts that irrational numbers exist. But in the end philosophers like Socrates won the debate and helped mathematics to flourish into many branches.

M. Serkan Kalaycıoğlu

Real Mathematics: Killer Numbers #3

Theodorus of Cyrene

City of Cyrene was one of the ancient Greek cities that were located in the Northern Africa at around 5th century BC. We are certain that Theodorus (465 BC – 398 BC) was born in Cyrene and he was a tutor of Plato. Theodorus probably met Socrates and lived in Athens for some time. Unfortunately we don’t know much about his life.

Although we know through his student Plato that Theodorus had made significant works on irrational numbers. During ancient times, even though they had number symbols called Attic (also  known as Herodianic symbols) Greek philosophers did not care about number symbols. According to their thinking numbers were just magnitudes of lines. Hence they used line segments to represent numbers.

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Pythagoreans thought that every number can be shown as a ratio of any other two rational numbers, and that is why they claimed that numbers are rational. Oddly enough a geometric property what we call Pythagoras Theorem contradicted Pythagoreans’ claims as an isosceles right triangle with 1 unit length sides would have a hypotenuse that is √2 unit length. Unfortunately for them √2 is not a rational number. In other words, it can’t be shown as a ratio of two rational numbers.

Theodorus was also a Pythagorean, but he proved that √2, √3, √5, … are all irrational. For some reason he stopped at √17.

In this article I’ll be talking about a work of Theodorus now called Spiral of Theodorus.

Construction of the Spiral of Theodorus

This lovely geometric shape, which is also called “Einstein Spiral”, “π Spiral” and “Square Root Spiral”, could be constructed by anyone who knows Pythagoras Theorem.

At first construct an isosceles right triangle with 1 unit length sides. Through Pythagoras Theorem we can calculate the length of the hypotenuse as √2.


In the next step, continue drawing a base with length 1 unit that is perpendicular to the hypotenuse of the previous triangle and construct a second triangle which would have hypotenuse with length √3.

As long as you continue the same process, you will be ending up with hypotenuses with lengths √4, √5, √6, √7 … Theodorus stopped at √17. Probably he ended his structure at 17 because that is the final triangle before overlapping starts.


It is obvious to the naked eyes that these triangles form a beautiful spiral, which is called Spiral of Theodorus.

Nothing like a spiral

What is so special about the Spiral of Theodorus?

  • All the lengths of the hypotenuses of the spiral, except the perfect square number lengths, are irrational.

    √1, √4, √9, √16, √25… are the only numbers that are rational.
  • If one continues to add triangles which would mean that the spiral is going to the infinity, no two hypotenuses overlap.

    Even if it looks too close, no two hypotenuses overlap in the Spiral of Theodorus.
  • The windings of the spiral would have length π between themselves as one adds infinitely many triangles.

    I have found 3,1 in the first 30 triangles. If I kept going for infinity, this number would approach to π.
  • The angle between two consecutive perfect square number hypotenuses would approach to 360/π as the spiral goes to infinity.

    360/π = 114,591559026… which is very close to what I draw. I was certainly lucky but if you draw this perfectly, as you approach to infinity, you would find exactly. 360/π.

Killer Curve

This is my favorite property of the Spiral of Theodorus: Cut off every single triangle from the spiral and align them on the x-y coordinate system.

If you connect the tipping points of every triangle, you would end up with the y=√x curve. I like calling irrational numbers as “killer numbers” because of the story of poor Hippasus. I think it would be suitable to call this curve The Killer Curve.

I also used a program called GeoGebra and find the following result.


One might wonder…

What would happen if you take the first triangle like the following?


What kinds of changes do you observe?

M. Serkan Kalaycıoğlu